arXiv:2607.14797 2026 Dynamics 1 ideas extracted · analyzed Aug 30, 2026
The paper gives a constructive small-perturbation principle for periodic unitary dynamics: if a base operator satisfies \(U_0^{\tau}=I\), then repeatedly applying a weakly perturbed \(\tau\)-step block for \(O(1/\beta)\) blocks converges to continuous-time evolution generated by a Hermitian effective Hamiltonian. The relevant first-order generator is especially informative inside degenerate eigenspaces, where projected perturbations determine how periodicity is broken. This can transfer to unitary RNNs and state-space layers by initializing an exactly periodic recurrent operator and learning a small Hermitian perturbation. The main hypothesis is that this preserves signal norms and improves long-horizon gradient behavior compared with unconstrained recurrent transitions.
Replace a generic recurrent transition by an exactly periodic unitary base transition plus a learnable weak Hermitian perturbation. The resulting \(\tau\)-step macro-dynamics approximates a continuous-time unitary flow, allowing the model to preserve signal norms while learning slowly varying long-range transformations.